The sequential closedness of \(\sigma\)-complete Boolean algebras of projections
type: Item
instance of: scholarly article
title: The sequential closedness of \(\sigma\)-complete Boolean algebras of projections
zbMATH Open document ID: 0968.47037
published in: Journal of Mathematical Analysis and Applications
publication date: 2001-09-03T00:00:00Z
review text: Let \(X\) be a locally convex Hausdorff space and let \(L(X)\) denote the space of all continuous linear operators of \(X\) into itself. If \(X\) is a separable Fréchet space, then any \(\sigma\)-complete (in the sense of Bade) Boolean algebra of projections is a strong operator closed subspace of \(L(X)\). It is no longer the case if \(X\) is non-separable, but then we can often replace closedness with sequential closedness. It is shown that under mild completeness assumptions on \(X\), one obtains the sequential closedness of a \(\sigma\)-complete atomic Boolean algebra of projections. The assumptions are satisfied if \(X\) is a Fréchet space. A counterexample is given for a non-atomic algebra.
Mathematics Subject Classification ID: 47L80
Mathematics Subject Classification ID: 06E10
Mathematics Subject Classification ID: 47A46
Mathematics Subject Classification ID: 46A04
zbMATH DE Number: 1031576
zbMATH Keywords: Boolean algebra
zbMATH Keywords: \(\sigma\)-completeness
zbMATH Keywords: projections
zbMATH Keywords: space of all continuous linear operators
zbMATH Keywords: separable Fréchet space
zbMATH Keywords: Boolean algebra of projections
zbMATH Keywords: strong operator closed subspace
zbMATH Keywords: sequential closedness
zbMATH Keywords: completeness
zbMATH Keywords: \(\sigma\)-complete atomic Boolean algebra
author: Werner J. Ricker
reviewed by: Stanisław Goldstein
MaRDI profile type: Publication
full work available at URL: jmaa.1997.5300
OpenAlex ID: W1967486372
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DOI: 10.1006/JMAA.1997.5300
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description: scientific article; zbMATH DE number 1031576